The analytical case studies of this thesis focus mainly on the findings associated with adapting the sensor selection process and the inverse method for the use on civil structures. Danai et al. 2011 reported work that has been done to adopt the direct method for the use on civil structures, results of which were included in the introduction as supplemental background information for this thesis.
Two structures to be used as the sample buildings, chosen for their prior usage on projects in the field of sensor selection and seismic research, were modeled using the structural analysis program OpenSees. The first structure is a simple, eight-story frame. The second is a slightly more complex, multi-bayed, nine-story structure. These structures were ideal for this project because they could be modified quickly for whatever purpose was required in the project but they were simple enough that a finite element analysis could be performed reasonably quickly. Additionally, the nine-story structure offered realistic characteristics of an actual structure, while the eight story building offered more optimal dynamic characteristics.
After a thorough investigation of this method with the purely analytical models, it was further tested on data from physical models. These physical models added the
variables of noise and modeling error which were not accounted for with the analytical models but would be present during real world testing. The research utilized reported results from previous research which had excited a structure both pre and post damage. A downside of using existing test data is that there was no control of the structure or how
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it was tested, the only usable information was the data that was reported by the researchers at the time the project was performed.
The physical model data was obtained from the Network for Earthquake Engineering Simulation (NEES) database of archived projects. The project that was chosen was a study originally preformed to investigate the interactions between a reinforced concrete frame and shear wall during strong earthquake excitations. The project was performed during May of 1979 at the University of Illinois at Urbana- Champaign. All available information about the project including publications, recorded data, drawings and photographs can be found in the NEES database at
http://nees.org/warehouse/project/1019. The particular project was chosen from the many available in the NEES database because it provided the dynamic responses of the test structures to various magnitudes of earthquake excitations both before damage occurred and subsequently after different stages of damage. Additionally, that project provided detailed drawings of the damage that occurred in the structures after each excitation which would be useful when verifying the results of the damage estimation procedure. Finally, the structures which the research team used were relatively simple models that were applicable for use in this current sensor selection project. The research team designed and built four tenth scale reinforced concrete structures which were outfitted with a full set of accelerometers, LVDTs and stain gages to record accelerations, displacements and forces at multiple locations on each floor.
The previous research provided the necessary damaged responses of the structures which could ultimately be used by the damage estimate procedure to approximate where the damage occurred. But since this current research is focused on minimizing the
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required sensors to correctly estimate damage the sensor selection process needed to be performed first. An analytical model which matched the physical model as closely as possible was developed in OpenSEES based on the specifications which were provided in the available publications. Using the newly created analytical models of the structures, preferred sensors could be chosen. Next, using only the responses generated by those preferred sensors on the physical model the damage in the structure was estimated. By verifying that the damage estimated by the damage estimation routine matched the actual damage that was reported in the previous research it could prove that the preferred sensor suite is a better fit to estimate damage than other sensors.
Eight Story Model
The eight story model (Figure 7) is a two-dimensional single-bay structure. The model characteristics were taken from a previous sensors optimizations study (Yuen et al. 2001) which investigated modal characteristics rather than the dynamic response. The original model from Yuen et al. was a simple mass-spring structure which was modified for this project to a simple frame structure. Although the shape of the model was updated the dynamic properties were kept the same, most importantly the stiffness to mass ratio of 1160 s-2 which creates a fundamental natural frequency in the structure of exactly 1.00 Hz
Each story in the structure consists of two massless columns of identical stiffness connected laterally by a rigid girder. The rigid girder was assigned a lumped mass which was equivalent for each story. The columns in the structure were fixed to the rigid girders with a moment connection at each end. The columns on the bottom story were fixed to
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the ground. The girders assigned infinite rigidity so that the stiffness of each story was independent of the stiffness any other story. Therefore, if the column stiffness of one story was decreased to simulate damage it would not change the column stiffness on any other story. The structural modeling was done in OpenSees using completely elastic beam elements for all of the structural members. Each of the columns was assigned a 3.05 m (10 ft) length and a modulus of elasticity of 200 GPa (29000 ksi). The girder span between the columns was 9.14 m (30 ft). The girders were assigned infinite rigidity by constraining the rotational degrees of freedom at each of the girder-column connections. Each column was assigned an area of 929 cm2 (144 in2) resulting in a moment of inertia of 71,925 cm4 (1728 in2). Because the columns were fixed at each end the resulting stiffness of each column was 30.47 MN/m (174 kips/in). Because each story had two columns, the effective stiffness of each story was twice that of a column, or 60.94 MN/m (348 kips/in). Each of the rigid beams was assigned a mass of 52,540 kg (0.3 kip-s2/in) and the columns were left massless. The resulting stiffness to mass ratio of each story was 1160 s-2 which produced a fundamental natural frequency in the structure of exactly 1.00 Hz.
Damage was introduced to this model as a reduction in the overall stiffness of an entire story. Therefore there were eight possible damage locations in this structure. The stiffness of a story was decreased by reducing the Modulus of Elastic (E) of both columns by a prescribed amount. The stiffness reduction is not cause specific and could represent distributed damage throughout a story or localized damage. Damage could be due to an extreme event or long term deterioration. The conceptual application is applied to a
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Figure 7 - The configuration of the eight story building model
situation where a building is instrumented and readings are taken for a baseline initial ``non-damaged'' condition (such as immediately after construction, but could be any point prior to measured damage) and a post damage reading. One possible option, and the most likely option for practical applications, was to excite the structure using an eccentric mass shaker such as those available as part of the Network of Earthquake Engineering Simulation facilities at UCLA (http://nees.ucla.edu/shakers.html). To simulate the eccentric mass shaker, lateral sinusoidal time series forcing functions were applied at individual nodes of the model. These forcing functions were applied at a specific
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magnitude, frequency and duration to produce a dynamic response in the structure. Another option was to use ground excitation such as an earthquake record. Although this option may not be practically feasible it was still beneficial to examine from a conceptual standpoint.
SAC Nine Story Model
The second model that was used as part of this research was a nine-story building frame with a basement. The structure that this model was taken from was developed as part of the SAC Phase II research initiative as described by Ohtori et al., 2005 as a reference for benchmark structural evaluations. While the eight story model used non- specific physical properties to achieve an optimal natural frequency, the physical properties of this model reflects those of material that would be used in an actual structure.
The SAC building consists of perimeter steel moment frames designed to meet seismic design requirements in Los Angeles, California. The model that was used in this project was a single North-South moment resisting frame of the SAC structure as shown in Figure 8. The modeled masses on the frame (one of two frames that would be provide for the North-South direction lateral resisting system) were 4.825 x 105 kg (2.75 kip- s2/in), 5.050 x 105 kg (2.88 kip-s2/in), 4.945 x 105 kg (2.81 kip-s2/in) and 5.350 x 105 kg (3.05 kip-s2/in) for the first floor, second floor, third to nineth floors and roof, respectively. Column splices shown in the figure were included as weighted averages of the above and below column properties within the story height, similarly to Ohtori et al., 2005. Column sizes and girder sizes vary throughout the height of the building,
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Figure 8 - The configuration of the SAC nine story building model
remaining similar at each floor level. Column and girder steel is modeled with a modulus of elasticity of 345 MPa (50.0 ksi), with non-linear material properties (strain hardening included), and non-linear geometric (P-Delta analysis) included in the analysis. Beam and
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column elements were modeled as fiber element W sections using nonlinear beam- column elements which allow for spread of plasticity in members. It is important to note that non-linear behavior was included in the model to capture behavior under full seismic loads being considered in other phases of the research project. However, all analysis for damage isolations reported in this paper is within the material elastic range of behavior. This was verified by comparing results to models with elastic material behavior and first- order analysis.
Similar to the eight-story model, this model was excited using forcing functions which were positioned at nodes throughout the structure to simulate eccentric mass shakers. Ground motion was also used to simulate an earthquake event, but in a limited capacity as it would not be practical for real world applications.
Sensor Selection with the Inverse Method
The method of sensor selection that is used in this research relies on the parameter signature matrix to determine the sensor suite. Since the influence matrix is found using the direct method it will therefore be related to the direct method. If a sensor suite was to be chosen for the direct method, the suite that provided identifiability throughout the entire structure should result in the highest likelihood of the direct method isolating damage. However, the inverse method itself is not directly related to the parameter signature matrix. Although the suite is the group of sensors which are most influenced by damage, the inverse method does not necessarily require exactly that suite in order to work such as the direct method would. The preferred sensors are more of a suggestion
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when it pertains to the inverse method rather than a requirement as with the direct method.
The inverse method may be able to estimate damage using fewer sensors than is suggested by a preferred suite or it may require more. The reason that this can occur is because the parameter signatures, which are used to create the parameter signature matrix, are created using only the most unique characteristics in the output sensitivities. If two sensors record responses with similar output sensitivities those output sensitivities will mask each other because they are not unique and therefore would not show up in the signatures. The direct method requires the unique output sensitivities in order to work which is why it is so dependent on the influence matrix, but the inverse method can work with even the small and repetitive changes that may not show up as unique signatures. However, this doesn’t mean that the parameter signature matrix is not important to the inverse method. It is intuitive to think that more unique signatures will allow the inverse method to converge to the correct parameters quicker and more accurately. It can also be assumed that sensors with unique changes must also have smaller changes which did not show up as a signature but can still help the inverse method converge. It is the primary purpose of this research to determine the validity of these assumptions and determine the usefulness of the parameter signature matrix when choosing a sensor suite for use with the inverse method.
Additional Cases
Supplemental case studies were also performed in this project to investigate the many factors that can affect the outcomes of the sensor selection and damage estimation
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procedures. The investigation of these factors was to determine valuable insight into the feasibility of implementing these procedures for real world applications. These factors include how and where the structure is excited in order to create the dynamic response, the use of transient versus steady state responses and the severity of damage.
Excitation Functions
A wide range of forcing functions were evaluated in order to determine which forcing functions were optimal for both the structure and the estimation procedures. Included in these forcing functions were simple sinusoidal waves with a constant frequency, sinusoidal sweeps over a wide frequency range, impulses and a number of earthquake records. Furthermore, the forcing functions could be applied to ground motion or at any of the floors in the structural models.
For the sensor selection or damage estimation procedures to work properly, an identical excitation function must be used to obtain the undamaged and the damaged responses. The excitation functions that can be applied to a computer model are nearly unlimited, but they are limited in application to an actual structure. The placement of a vibration generator in a structure is limited by the space and accessibility in the structure, while the frequency and amplitude of the vibrations are limited to the capacity of eccentric mass shaker. Secondly, ground motion can be eliminated as a means of exciting the damaged structure because it is impossible to predict the time or response of the earthquake before it occurs or obtain identical excitations at different points in time. Therefore, the most practical excitation configuration would be a sinusoidal forcing function. In an existing building the roof would be the most accessible location to place a
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vibration generator though would most likely need a crane in order to lift it into position. However, as the research has shown, this is not always the ideal configurations for these models. If a structure were designed to accommodate a lifelong monitoring plan a wide range of excitation source locations could be designed for implementation.
It can be shown that the preferred sensor suite for a structure varies depending on the excitation type. By changing the placement of the forcing function as well as other characteristics such as its frequency, the parameter signature matrix change, and therefore so does the sensor suite. It has also been determined that some excitation functions create a more preferred sensor suite than others. Table 8 shows two parameter signature matrices, both created with a dominance factor of 1.5 for an excitation at the second floor and an excitation at the roof of the nine story model. It is shown in the table that a full range of damage detection (unique signatures) requires only sensors at floor three, six and seven while using a second floor excitation. However sensors at three, four, six and eight are required to detect damage while using roof excitations. It is also possible to select other combinations of floors which would also provide a sensor suite.
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Table 8 - Sensor Suites for Excitations at the Second Floor and the Roof
Another determination which needs to be considered when performing damage estimation is what type of response will be used. From a practical stand-point the steady state response is much easier to acquire in the field with confidence of repeatability. However, because of the uniqueness of the transient response, it provides the estimation procedures with data which is better for isolating and estimating damage. The problem with the transient response is the difficulty of recording a consistent transient response from the actual structure that can closely match the response of the computer model.
A transient response has a seemingly random and non-repetitive pattern of motion while the steady state response is consistent and unchanging. Therefore, the output sensitivity that occurs in the transient region is also non-repetitive while the output sensitivity from the steady state region of a sine wave falls into a cyclical pattern. A transient response will yield more unique parameter signature which provides the sensors with better identifiability. This does not mean that a steady state response will not
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provide identifiability, and therefore could still be used. The reason that a transient response is so hard to properly acquire is because of the variability of the vibration generators. On a computer model a force can be applied instantly at any amplitude and in any direction, however a vibration generator does not have those capabilities. They cannot apply force instantaneously and must accelerate from a static position before being able to apply the desired amount of force. Applying a forcing function to the simulated model that can exactly match the forces created by a vibration generator in the initial instances is very difficult but critical to the procedure. Steady state is preferred because a response will always reach a constant steady state no matter how different the initial stages of the forcing functions began. However, it would also be considered transient to capture data as a temporal section of data during which the excitation is changing, such as a transition from one steady state excitation to another.
The frequency of the forcing function also has an effect on the optimum sensor suite. However, precautions must be taken not to excite a structure at or near a modal frequency which could incite resonance and possibly cause more damage in the way of plastic deformations in the structure. To prevent this, the model structures were only excited at frequencies which occurred at the median of any two adjacent modal frequencies i.e. the average frequency of the first and second modes, the second and third modes etc. In an attempt to eliminate the need to run analyses at each individual frequency a sine sweep function was also modeled. The function continuously increased frequency as it gradually swept through each desired frequency range in one analysis. The sine sweep provided some valuable information. It was determined that the frequencies beyond the third mode were so high that the finite element analysis could not
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get an accurate representation of the dynamic response. The analysis used time steps of .01 s while the modal periods associated with higher modes became just as small or even smaller. In order to achieve an accurate record of the response the time steps in the finite element analysis would need to be refined greatly which was too costly in a